Past exam of the physics B course of the University of Cambridge / 2012 / Paper 1 / Section A / A4 / Solution

The Stefan–Boltzmann law gives solar luminosity $L_{\odot }=4\pi R_{\odot }^2\sigma T_{\odot }^4$. At orbital distance $r$, the incident irradiance is $\sigma T_{\odot }^4{R}_{\odot }^2/r^2$. A spherical absorber intercepts a disc of area $\pi R_M^2$ but radiates over $4\pi R_M^2$. Thus the uniform-temperature radiative equilibrium estimate isThe planetary radius cancels. This is a black-body radiation estimate with zero reflected fraction and complete redistribution of absorbed heat. More generally the balance fixes $⟨T^4{⟩}^{1/4}$, rather than the arithmetic mean of a nonuniform surface temperature. It excludes atmospheric heating and thermal lag; no measured surface temperature is inferred.